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Graphs and sign tests for squared functions greater or less than a number.

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Inequalities

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Vocabulary

Compete the table.
 Word Definition Quadratic Inequality ___________________________________________________ ___________ the values of x that make y equal to zero in a quadratic function Polynomial Inequality: ___________________________________________________ ___________ A function which may be expressed as a ratio of two polynomials, specified to be greater or less than a given value

To solve quadratic inequalitites, you can graph it and see visually where the inequality is true. Without graphing, you can follow these steps:

1. Set up the inequality in the form    (or  \begin{align*}p(x)<0, p(x)\le0,p(x)\ge0\end{align*} )
2. Find the solutions to the equation \begin{align*}p(x)=0\end{align*} .
3. Divide the number line into intervals based on the solutions to \begin{align*}p(x)=0\end{align*} .
4. Use test points to find solution sets to the equation.

Example

Find the solution set for the equation .

First, factor:

\begin{align*}x^2 + 2x - 8 = 0\end{align*}

\begin{align*}(x + 4)(x - 2) = 0\end{align*}

Then create the intervals:

\begin{align*}(-\infty,-4) | (-4,2) | (2,\infty)\end{align*}

Finally, make a chart.

Interval Test Point Is \begin{align*}x^{2}+2x-8\end{align*} positive or negative? Part of Solution set?
\begin{align*}(-\infty,-4)\end{align*} \begin{align*}-5\end{align*} \begin{align*}+\end{align*} \begin{align*}yes\end{align*}
\begin{align*}(-4, 2)\end{align*} \begin{align*}1\end{align*} \begin{align*}-\end{align*} \begin{align*}no\end{align*}
\begin{align*}(2,+\infty)\end{align*} \begin{align*}3\end{align*} \begin{align*}+\end{align*} \begin{align*}yes\end{align*}

if and only if \begin{align*}x < -4\end{align*} and  .

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Polynomial and Rational Inequalities

In polynomial and rational inequalities, the four basic steps are the same as in polynomial inequalities. For rational inequalities, however, you must also account for possible sign changes at vertical asymptotes or a break in the graph.

Add the vertical asymtotes to your intervals, and create the same chart as you did for quadratic inequalities.

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Practice

1) Find the solution set of the inequality \begin{align*}x^2 \leq 36\end{align*}

2) Find the solution set:

3) Find the solution set:

4) Graph the solution set: \begin{align*}(x - 3)(x + 4) \geq 0\end{align*}

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Find the solution set of the following inequalities without using a calculator. Display the solution set on the number line.
1. \begin{align*}x^{2}+2x-3\le0\end{align*}
2. \begin{align*}-6x^{2}-13x+5\ge0\end{align*}
3. \begin{align*}\frac{1-x}{x}<1\end{align*}
4.  \begin{align*}\frac{x^4}{4} - x^2 <0\end{align*}
5. \begin{align*}4x^3 - 8x^2 - x + 2 \geq 0 \end{align*}
Solve the following inequalities:
1. \begin{align*}\frac{n^3 - 2n^2 - n + 2}{n^3 + 3n^2 + 4n + 12} < 0 \end{align*}
2. \begin{align*}\frac{n^3 + 3n^2 - 4n - 12}{n^3 - 5n^2 + 4n - 20} \leq 0 \end{align*}
3. \begin{align*}\frac{2n^3 + 5n^2 - 18n - 45}{3n^3 - n^2 +27n - 9} \geq 0 \end{align*}

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